Optimal. Leaf size=38 \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b^2}-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2} \]
[Out]
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Rubi [A] time = 0.0657783, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b^2}-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.91357, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0316473, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{5/2} \left (5 b x^2-2 a\right )}{35 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-5\,b{x}^{2}+2\,a}{35\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234097, size = 61, normalized size = 1.61 \[ \frac{{\left (5 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} + a^{2} b x^{2} - 2 \, a^{3}\right )} \sqrt{b x^{2} + a}}{35 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.44814, size = 85, normalized size = 2.24 \[ \begin{cases} - \frac{2 a^{3} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b} + \frac{8 a x^{4} \sqrt{a + b x^{2}}}{35} + \frac{b x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212545, size = 105, normalized size = 2.76 \[ \frac{\frac{7 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}}{b}}{105 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^3,x, algorithm="giac")
[Out]